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Vol. 11, No. 2, 2016, 95-100

ISSN: 2320 –3242 (P), 2320 –3250 (online) Published on 22 December 2016

www.researchmathsci.org

DOI: http://dx.doi.org/10.22457/ijfma.v11n2a5

95

International Journal of

On Corona Product of Two Fuzzy Graphs

Özge Çolakoğlu Havare and Hamza Menken

Department of Mathematics, Science and Arts Faculty, Mersin University Mersin, Turkey, 33343

E-mail: ozgecolakoglu@mersin.edu.tr; hmenken@mersin.edu.tr

Received 14 December 2016; accepted 21 December 2016

Abstract. Corona product of two fuzzy graphs which is analogous to the concept corona

product operation in crisp graph theory is defined. The degree of an edge in corona product of fuzzy graphs is obtained. Also, the degree of an edge in fuzzy graph formed by this operation in terms of the degree of edges in the given fuzzy graphs in some particular cases is found. Moreover, it is proved that corona product of two fuzzy graphs is effective when two fuzzy graphs is effective fuzzy graphs.

Keywords: Corona product; degree of an edge; effective fuzzy graph AMS Mathematics Subject Classification (2010): 03E72, 05C72

1. Introduction

It was Rosenfeld who considered fuzzy relations on fuzzy sets and developed the theory of fuzzy graphs in 1975 [10]. Later on, Bhattacharya gave some remarks on fuzzy graphs [1]. The operations of union, join, Cartesian product and composition on two fuzzy graphs were defined by Moderson and Peng [4]. The degree of a vertex in fuzzy graphs which are obtained from two given fuzzy graphs using these operations were discussed by Nagoorgani and Radha [6]. Radha and Kumaravel introduced the concept of degree of an edge and total degree of an edge in fuzzy graphs [8] and studied about the degree of an edge in fuzzy graphs which are obtained from two given fuzzy graphs using the operations of union and join [9].

In this paper, we have introduced the concept of corona product of fuzzy graphs, which are analogous to the concept corona product in crisp graph theory. We study about the degree of an edge in fuzzy graph which are obtained from two fuzzy graphs using corona product operation. The degree of an edge in the corona product of two fuzzy graphs is obtained in some particular case. Moreover, it is proved that corona product of two effective fuzzy graphs is an effective fuzzy graph.

Let V be a nonempty set. A fuzzy graph is a pair of functions G: ( , )σ µ where σ is a fuzzy subset of v and

µ

is a symmetric fuzzy relation on σ:V →[0,1] and

:V V [0,1]

µ × → such that µ

( )

u v, ≤σ

( ) ( )

u ∧σ v for all u v, in V [5]. The underlying crisp graph of G: ( , )σ µ is denoted by G*: (V, E) where E⊆ ×V V . µ

( )

u v, >0 for

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96

Throughout this paper we assume that

µ

is reflexive and need not consider loops. Note that Gi: (σ µi, i) denote fuzzy graphs with underlying crisp graphs

*

: (V , E ), i 1, 2

i i i

G = with Vi =pi, i=1, 2. Also, the underlying set V is assumed to be finite and

σ

can be chosen in any manner so as to satisfy the definition of a fuzzy graph in all the examples and all these properties are satisfied for all fuzzy graphs except null graphs. We shall denote the edge between two vertices u and v by uv.

In [6], the degree of a vertex u in G is defined by

( )

( )

( )

G

u v uv E

d u µ uv µ uv

≠ ∈

=

=

(1.1)

By Nagoorgani and Ahamed in [8], the order of a fuzzy graph G is defined by

( )

(u)

u V

O G σ

=

. (1.2) The union of two fuzzy graphs G1: (

σ µ

1, 1)and G2: (

σ µ

2, 2) is defined as a fuzzy graph G=G1∪G : (2

σ

1

σ µ

2, 1

µ

2) on G*: (V, )E where V = ∪V1 V2 and

1 2

E=EE with

( )

( )

( )

( )

( )

1 1 2

1 2 2 2 1

1 2 1 2

,

( ) ,

,

u u V V

u u u V V

u u u V V

σ

σ σ

σ

σ

σ

∈ −

 

∪ = ∈ −

∨ ∈ ∩

and

(

)( )

( )

( )

( )

( )

1 1 2

1 2 2 2 1

1 2 1 2

,

,

,

uv uv E E

uv uv uv E E

uv uv uv E E

µ

µ

µ

µ

µ

µ

∈ −

 

∪ = ∈ −

∨ ∈ ∩

.

Assume that V1∩ = ∅V2 . The join of G1 and G2 is defined as a fuzzy graph

(

)

1 2: 1 2, 1 2

G=G +G

σ σ µ µ

+ + on G*:

(

V E,

)

where V = ∪V1 V2 and

1 2 '

E= ∪E EE where E' is the set of all edges joining vertices of V1 with vertices of 2

V , with

(

σ σ

1+ 2

)( ) (

u =

σ σ

1∪ 2

)( )

u for all u∈ ∪V1 V2 and

(

)( ) (

1

( )

2

)( )

( )

1 2

1 2

1 2

,

.

, '

uv uv E E

uv

u v uv E

µ µ

µ µ

σ σ

∪ ∈ ∪



+ =

∧ ∈



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97

( )

( )

( )

2

( )

( )

( )

G G G

uw E wv E

w v w u

d uv d u d v µ uv µ uw µ wv

∈ ∈ ≠ ≠

= + − =

+

(1.3)

2. Degree of an edge in Corona product

In this section, we give the definition of corona product operation and calculated degree of an edge of fuzzy graphs that are obtained by this operation.

The corona of two graphs is defined in [3] and there have been some results on the corona of two graphs [2]. The corona product of two graphs G and H; denoted by

G H ; is the graph obtained by taking one copy of G of order n and n copies of H, and

then joining by an edge the i-th vertex of G to every vertex in the i-th copy of H. The corona product is neither associative nor commutative. Let σi be a fuzzy subset of Vi and

let µi be a fuzzy subset of Ei, i=1,2. Using definition of join and union, define the fuzzy

subset σ σ1 2 of V and µ µ1 2 of E as follows:

(

σ σ1 2

)( ) (

u = σ σ1∪ ∪ ∪2 ... σ2

)( )

u (union ofσ1and p1timesσ2)∀ ∈u V (2.1)

(

)( ) (

1

( )

2

( )

2

)( )

1 1 2

1 2

1 2

... , (union of times ) '

, '

uv and p uv E E

uv

u v uv E

µ µ µ µ µ

µ µ

σ σ

∪ ∪ ∪ ∈ −



=

∧ ∈



(2.2)

Where E is the set of all edges joining by an edge the ' the i-th vertex of G to every vertex in the i-th copy of H.

Theorem 2.1. Let G=G G1 2. For any uvE, 1) If uv∈ −E E' then

( )

1

( )

( )

2

( )

( )

( )

( )

2

( )

( )

( )

( )

2

1 1

1 2 1 2 1

1 2 1 2 2

,

,

G

w V w V

G

G

w V w V

d uv u w v w uv E

d uv

d uv w u w v uv E

σ

σ

σ

σ

σ

σ

σ

σ

∈ ∈

∈ ∈

+ +

=

+ ∧ + ∧ ∈

 

(2.3)

2) If uvE' then

( )

1

( )

2

( )

1

( )

2

( )

'

G G G

uw E w v

d uv d u d v σ u σ w

∈ ≠

= + +

∧ (2.4)

Proof: By (1.3), we have

( )

( )

( )

( )

' ' ' '

(uv)

G

uw E E wv E E uw E wv E

w v w u w v w u

d

µ

uw

µ

wv

µ

uw

µ

wv

∈ − ∈ − ∈ ∈

≠ ≠ ≠ ≠

=

+

+

+

(2.5)

Assume that uv∈ −E E' with 1

uvE . Using (2.2) in (2.5) we get

( )

( )

( )

( )

( )

( )

( )

1 1

2 2

1 2 2 1

' '

G

uw E wv E uw E wv E

w V w V

w v w u

d uv µ uw µ wv σ u σ w σ w σ v

∈ ∈ ∈ ∈

∈ ∈

≠ ≠

=

+

+

∧ +

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98

Now, let uvE' withu V v V1, ∈ 2. Using (2.2) in equation (2.5) we see that

( )

( )

( )

( )

( )

( )

( )

1 2

1 2 1 2

' '

G

w V w V uw E wv E

w v w u

d uv µ uw µ wv σ u σ w σ w σ v

∈ ∈ ∈ ∈

≠ ≠

=

+

+

∧ +

From definition of corona product, if uvE',u V v V1, ∈ 2 and wu then w V1.

As a conclusion, by (1.1) we obtain equation (2.4). Thus, we complete proof of the theorem. □

In the following theorems, we find the degree of uv in

G

in terms of those in

k

G for k=1, 2 in some particular cases.

Nagoorgani and Radha in [5] defined the relation σ σ12 means that

( )

( )

1 u 2 v

σ ≥σ , for every u V1 and for every v V2, where σi is a fuzzy subset of , 1, 2.

i

V i=

Theorem 2.2. Let G=G1G2. For σ2≥σ1 the following equalities holds:

1) If uv∈ −E E' then

( )

( )

(

( )

( )

)

( )

( )

1

2

2 1 1 1

1 1 2

,

2 ,

G G

G

d uv p u v uv E

d uv

d uv w w E and uv E

σ

σ

σ

+ +

=

+ ∈ ∈

 .

2) If uvE' with u V1, v V2 then

( )

1

( )

2

( ) (

2 1

) ( )

1

G G G

d uv =d u +d v + p

σ

u

Proof: We haveσ2σ1. Let any uv∈ −E E'. From equation (2.3) for

1

uvE we have

( )

1

( )

( )

( )

2 2

1 1

G G

w V w V

d uv d uv

σ

u

σ

v

∈ ∈

= +

+

.

Recall that Vi = p ii, =1, 2. Hence, we have

( )

1

( )

2

(

1

( )

1

( )

)

G G

d uv =d uv +p

σ

u +

σ

v . Now, for uvE2, from equation (2.3) we have

( )

2

( )

( )

( )

1 1

1 1

G G

w V w V

d uv d uv

σ

w

σ

w

∈ ∈

= +

+

Therefore we obtain Theorem 2.2 (1).

Using conditions of the Theorem 2.2 in equation (2.4), we get that

( )

1

( )

2

( )

( )

2

1

G G G

w V w v

d uv d u d v σ u

∈ ≠

= + +

Now, using equation (1.2) and Vi =p ii, =1, 2, we obtain that

( )

( )

( ) (

) ( )

1 2 2 1 1

G G G

d uv =d u +d v + p

σ

u

Theorem 2.3. Let G=G1G2. For σ σ1≥ 2 the following equalities holds:

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99

( )

1

( )

( )

( )

( )

( )

2

2 1

2 2 2

2 ,

,

G G

G

d uv O G uv E

d uv

d uv

σ

u

σ

v uv E

+ ∈



=

+ + ∈

 .

2) If uvE' with u V∈ 1, v V∈ 2 then

( )

1

( )

2

( ) ( )

2 2

( )

G G G

d uv =d u +d v +O G

σ

v

Proof: We have σ σ1≥ 2. Let any

'

uv∈ −E E. In similar a way, by equation (2.3) for 1

uvE we have

( )

1

( )

( )

( )

2 2

2 2

G G

w V w V

d uv d uv

σ

w

σ

w

∈ ∈

= +

+

.

By equation (1.2), we get

( )

1

( )

2

( )

2

G G

d uv =d uv + O G . For uvE2, from equation (2.3),

( )

2

( )

( )

( )

1 1

2 2

G G

w V w V

d uv d uv

σ

u

σ

v

∈ ∈

= +

+

From definition of corona product, we obtain Theorem 2.3 (1). For any uvE' with

1, 2

u Vv V∈ . Using (2.4), we get

( )

1

( )

2

( )

( )

2

2

G G G

w V w v

d uv d u d v σ w

∈ ≠

= + +

By equation (1.2), we obtain Theorem 2.3 (2). □

Theorem 2.4. The corona product of two effective fuzzy graphs is an effective fuzzy graph.

Proof: Let G1

(

σ µ

1, 1

)

and G2 =

(

σ µ

2, 2

)

be effective fuzzy graphs. Then

( )

( )

( )

1 u v1 1 1 u1 1 v1

µ

=

σ

σ

for uvE1 and

µ

2

(

u v2 2

)

=

σ

2

( )

u2

σ

2

( )

v2 for

2

uvE . Let G=G G1 2. By (2.2), the fuzzy subset

µ µ

1 2 of E is

(

)( )

(

(

( )

( )

)

(

( )

( )

)

)

( )

( )

( )

1 1 1 1 2 2 2 2 2

1 2

1 2

... , '

, '

u v u v uv uv E E

uv

u v uv E

σ σ σ σ µ

µ µ

σ σ

∪ ∪ ∈ −

=

∧ ∈



Thus, the proof of the theorem is completed. □

REFERENCES

1. A.Bhattacharya, Some remarks on fuzzy graphs, Pattern Recognition Letters, 9 (1987) 159-162.

2. R.Frucht and F.Harary, On the corona two graphs, Aequationes Math., 4 (1970) 322-325.

3. F.Hdrary, Graph Theory, Addition-Wesley pub. Co., London, 1969.

4. J.N.Mordeson and C.S.Peng, Operation on fuzzy graphs, Information Sciences, 79 (1994) 159-170.

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100

6. A.Nagoorgani and K.Radha, The degree of a vertex in some fuzzy graphs,

International Journal of Algorithms Computing and Mathematics, 2 (2009) 107-116.

7. A.Nagoorgani and M.Basheer Ahamed, Order and size in fuzzy graph, Bulletin of

Pure and Applied Sciences, 22E (1) (2003) 145-148.

8. K.Radha and N.Kumaravel, The degree of an edge in Cartesian product and composition of two fuzzy graphs, Intern. J. Applied Mathematics & Statistical

Sciences, 2(2) (2013) 65-78.

9. K.Radha and N.Kumaravel, The degree of an edge in union and join of two fuzzy graphs, International Journal of Fuzzy Mathematical Archive, 4(1) (2014) 8-19. 10. A.Roseneld, fuzzy graphs, in: L.A.Zadeh, K.S. Fu. K .Tanaka, M.Shimura (Eds),

Fuzzy sets and Their Applications to Cognitive and Decision Processes, Academic

References

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